be exhausted, the other containing air, and if the
temperatures be equal, evaporation will go on until the pressure of the
vapour in the exhausted vessel is equal to its _partial_ pressure in the
other vessel, notwithstanding the fact that the _total_ pressure in the
latter vessel is greater by the pressure of the air.

To separate mixed gases by liquefaction, they must be compressed and
cooled till one separates in the form of a liquid. If no changes are
to take place outside the system, the separate components must be
allowed to expand until the work of expansion is equal to the work of
compression, and the heat given out in compression is reabsorbed in
expansion. The process may be made as nearly reversible as we like by
performing the operations so slowly that the substances are
practically in a state of equilibrium at every stage. This is a
consequence of an important axiom in thermodynamics according to which
"any small change in the neighbourhood of a state of equilibrium is to
a first approximation reversible."

Suppose now that at any stage of the compression the partial pressures
of the two gases are p1 and p2, and that the volume is changed from V
to V - dV. The work of compression is (p1 + p2)dV, and this work will
be restored at the corresponding stage if each of the separated gases
increases in volume from V - dV to V. The ultimate state of the
separated gases will thus be one in which each gas occupies the volume
V originally occupied by the mixture.

We may now obtain an estimate of the amount of energy rendered
unavailable by diffusion. We suppose two gases occupying volumes V1
and V2 at equal pressure p to mix by diffusion, so that the final
volume is V1 + V2. Then if before mixing each gas had been allowed to
expand till its volume was V1 + V2, work would have been done in the
expansion, and the gases could still have been mixed by a reversal of
the process above described. In the actual diffusion this work of
expansion is lost, and represents energy rendered unavailable at the
temperature at which diffusion takes place. When divided by that
temperature the quotient gives the increase of entropy. Thus the
irreversible processes, and, in particular, the entropy changes
associated with diffusion of two gases at uniform pressure, are the
same as would take place if each of the gases in turn were to expand
by rushing into a vacuum, till it